Communications in Algebra 13(1), (1985) 1

Transcription

1 Communications in Algebra 13(1), (1985) 1 CROSSED PRODUCTS AND BLOCKS WITH NORMAL DEFECT GROUPS Burkard Külshammer Department of Mathematics University of Dortmund, Dortmund Blocks having a normal defect group D are fairly well understood. R. Brauer [3] reduced the analysis of their block idempotents to blocks of defect zero in the quotient of the centralizer of D by the center of D. A reduction for the ordinary and modular characters in such blocks to the twisted group algebra of a naturally defined group having D as normal Sylow p-subgroup was given by Reynolds [10]. Later Dade [4] extended Brauer s work to study the behavior of arbitrary block idempotents with respect to normal subgroups. Here we use Dade s methods to generalize Reynolds theorem. Since our reduction produces a crossed product it seems natural to start with crossed products, too. So we work in this slightly more general context. As a corollary we get the following version of Reynolds result: A. THEOREM. Let G be a finite group and R a complete discrete valuation ring with residue class field F of prime characteristic p. Let B E be a block of the group algebra RG having a normal defect group D, and choose a block b e of RC G (D) with Ee 0. Denote by G(b) the set of all elements g G such that e g = e, and assume that F is a splitting field for Zb. Then DC G (D)/C G (D) is a normal Sylow p-subgroup of G(b)/C G (D), and B is isomorphic as an R-algebra to S R R γ DH for some central separable R-algebra S and some twisted group algebra R γ DH of DH over R where H denotes a complement of DC G (D)/C G (D) in G(b)/DC G (D) and DH the semidirect product of D with H. Let us fix some notation. Throughout the following, R denotes a complete discrete valuation ring with residue class field F. (We also allow the case that R is a field.) All R-algebras are assumed to have an identity and to be free R-modules of finite rank. For subsets X, Y of an R-algebra A we denote by C X (Y ) := {x X xy = yx for y Y } the centralizer of Y in X. In particular, C A (A) = ZA is the center of A. We denote by JA the Jacobson radical and by UA the group of units of A. The group Aut A of all R-algebra automorphisms of A has as normal subgroup the group Inn A consisting of all inner automorphisms A A, a au = a u, for u UA. For a natural number n, Mat(n, A) will be the R-algebra of all n n-matrices with coefficients in A. We denote blocks of A by B E where E is the block idempotent and B = EA the block ideal. If G is a group acting on the center of an R-algebra A we denote by G(B) := {g G E g = E} the stabilizer of a block B E of A. 1. CROSSED PRODUCTS We us the approach given by Dade in his elegant paper [5]. Proofs of the elementary facts on crossed products are left to the reader. Some of them may

2 Communications in Algebra 13(1), (1985) 2 be found in [4,5]. Let G be a finite group, and let A be an R-algebra with fixed decomposition A = A g where A g is an R-submodule of A containing a unit of A such that g G A g A h A gh for all g, h G. Then A is called a crossed product of G over A 1, and for g G the R-submodule A g of A is called the g-component of A. It is easy to see that A 1 is an R-subalgebra of A containing the identity of A. For any unit u g of A in A g we have A g = A 1 u g = u g A 1. Therefore A g A h = A gh for g, h G. The structure of A is usually described by an automorphism system and a factor system. A map α : G Aut A 1 is called an automorphism system for A if, for any g G, there is a unit u g of A in A g such that x α(g) = g xu g for x A 1. If α is an automorphism system for A then all automorphism systems are given by the maps ια : G Aut A 1, g ι(g)α(g), where ι : G Inn A 1 is an arbitrary map. Therefore the induced map G Aut A 1 /Inn A 1, g α(g)inn A 1 is independent of the choice of the automorphism system α and a homomorphism of groups. The usual problem is that α itself need not be a homomorphism. It is possible, however, to define an action of G on C A (A 1 ) by setting c g := g cu g for elements g G, c C A (A 1 ) and any unit u g of A in A g. Since for different units u g, v g of A in A g the element vg 1 u g lies in UA 1 this action is in fact well-defined. By restriction we get an action of G on ZA 1. It can be shown that the map α : G Aut A 1 determines the isomorphism type of A (as a crossed product of G over A 1 ) up to an element in the second cohomology group H 2 (G, UZA 1 ) but we shall not go into this here in detail. For any subgroup H of G we set A H := A h. Then A H is a crossed product of H over A 1. Furthermore, for any normal subgroup N of G, A is a crossed product of G/N over A N with gn-component A gn for g G. n N Moreover, if I is an ideal of A containing (JR)A such that I = I A g then A/I is a crossed product of G over the F -algebra A 1 + I/I = A 1 /A 1 I with g-component (A/I) g = A g + I/I for g G. Now let H and K be subgroups of G with K H, and choose a transversal T for K in H. Then, for any element c C A (A K ), the element c t is easily seen to be in C A (A H ) and independent of the choice of T. We denote this element by τk H(c). Then we get a relative trace map τ K H : C A(A K ) C A (A H ) which commutes with multiplication by elements in C A (A H ). It is obvious that τk H will have the usual properties of a relative trace map (cf. [7]). The following elementary result is one of the main tools in our analysis. B. LEMMA. Let A be a crossed product of G over A 1 with automorphism system α. Let S be an R-subalgebra of A 1 such that for any element g G there is a unit u(g) in A 1 satisfying s α(g) = u(g)su(g) 1 for any s S. Then the following holds: (i) C A (S) is a crossed product of G over C A1 (S). (ii) The map α : G Aut A 1 with x α (g) = x α(g)u(g) for x A 1 is an automorphism system for A such that its restriction G Aut C A1 (S) is an automorphism system for C A (S). (iii) A = A 1 C A (S) = C A (S)A 1. h H g G t T

3 Communications in Algebra 13(1), (1985) 3 Proof. We choose units u g of A in A g such that x α(g) = g xu g for x A 1 and g G. Then the elements u g u(g) are units of A in A g centralizing S. Since S is contained in A 1 we get C A (S) = g G C A (S) A g = g G C A1 (S)u g u(g), and A = g G A 1 u g u(g) = g G u g u(g)a 1. Therefore A = A 1 C A (S) = C A (S)A 1, and the result follows. It is easily seen that the hypothesis of lemma B does depend on the choice of the automorphism system. A first application is the following: C. THEOREM. Let A be a crossed product of G over A 1 with automorphism system α, and set N := {g G α(g) Inn A 1 }. Then the following holds: (i) N is a normal subgroup of G. (ii) The For any transversal T for N in G the map α : G/N Aut A N satisfying (xc) α (gn) = x α(g) c g for elements x A 1, c C AN (A 1 ), g T is an automorphism system for the crossed product A of G/N over A N. Proof. N is the kernel of the homomorphism G Aut A 1 /Inn A 1, g α(g)inn A 1, so it is a normal subgroup of G. Then A may be considered as a crossed product of G/N over A N, and A N is a crossed product of N over A 1. We may apply lemma B with N in place of G and A 1 in place of S. This gives A N = A 1 C AN (A 1 ) = C AN (A 1 )A 1, and C AN (A 1 ) is a crossed product of N over ZA 1. To prove (ii) it is enough to show that A 1 ZA1 C AN (A 1 ) and A N are free R-modules of equal rank. But C AN (A 1 ) is free of rank N as a module over ZA 1, so A 1 ZA1 C AN (A 1 ) is free of rank N over A 1. Since the same is true for A N, (ii) follows. (iv) is an immediate consequence of lemma B. Now choose a unit u g of A in A g for g T such that x ug = x α(g) for x A 1. Then A = A N u g, and (xc) ug = x α(g) c g for x A 1, c C AN (A 1 ), g T. g T This implies (v). In applications of theorem C it may be difficult to determine the structure of C AN (A 1 ). We introduce further restrictions in order to simplify the situation. Azumaya [1] has shown that any R-algebra A such that A/JA is a separable F -algebra contains an R-separable subalgebra S such that A = S + JA. Furthermore, S is uniquely determined up to inner automorphisms of A. Such a subalgebra S is called an inertial subalgebra of A. For elementary facts on separable algebras we refer to [2,6]. The following lemma will later be applied to a block of C AN (A 1 ). D. LEMMA. Let A be an R-algebra and Z a local subalgebra of ZA such that A is free over Z and A/(JZ)A is a separable F -algebra. Then the following holds:

4 Communications in Algebra 13(1), (1985) 4 (i) JA = (JZ)A, so A contains an inertial subalgebra. (ii) Z/JZ is a separable F -algebra, so Z contains one and only one inertial subalgebra T. (iii) T is a complete discrete valuation ring satisfying JT = (JR)T. (iv) T = S Z for any inertial subalgebra S of A, and the map Z T S A, z s zs, is an isomorphism of R-algebras. Proof. (i) follows immediately from our hypothesis. (ii) By [2, III.2.19] Z/JZ is a separable F -algebra, and A/(JZ)A is a separable Z/JZ-algebra. Then, by Azumaya s theorem, Z contains a unique inertial subalgebra T. (iii) Since T is separable over R we get JT = (JR)T. T is a local R-algebra since T is contained in Z. Since T is torsion-free as an R-module it is an integral domain. Therefore T is a discrete valuation ring and complete since the rank of T over R is finite. (iv) It is easy to see that the hypothesis of lemma D is also satisfied with T in place of R. Let S be a T -separable inertial subalgebra of A. Then T is also an R-separable inertial subalgebra of A, and T S Z. We choose a basis s 1,..., s n for S over T. Then s 1 + JS,..., s n + JS is a basis of S/JS = S/(JT )S over T/JT, and s 1 + JA,..., s n + JA is a basis of S + JA/JA = A/JA over T/JT = Z/JZ. This implies that s 1,..., s n is a basis for A over Z. In particular the map Z T S A, z s zs, is an isomorphism of R-algebras. We get T = S Z since s 1,..., s n are linearly independent over T and over S Z. Since property (iv) is invariant under conjugation with units the result follows. The following is a well-known property of central separable R-algebras. E. LEMMA. Let A be an arbitrary R-algebra and S a central separable subalgebra of A. Then the map C A (S) R S A, c s cs, is an isomorphism of R-algebras. Proof. See [2, III.4.3] We want to study the hypothesis of lemma B when S is a separable R- algebra. Therefore the following well-known lemma is useful. F. LEMMA. Let S be a separable R-algebra, A an arbitrary R-algebra and σ, σ : S A homomorphisms of R-algebras such that s σ + JA = s σ + JA for s S. Then there is an element x JA such that (1 x) 1 s σ (1 x) = s σ for s S. Proof. The maps σ and σ turn JA into an S-bimodule by sxt := s σ xt σ for elements x JA, s, t S. Then the map S JA, s s σ s σ, is easily seen to be a derivation. By the cohomological characterization of separable algebras any derivation of S is inner ([6, p. 76]). So there is an element x JA such that s σ s σ = s σ x = xs σ for s S. Then s σ (1 x) = (1 x)s σ, and the result follows since 1 x is a unit in A.

5 Communications in Algebra 13(1), (1985) 5 More can be said when S is even central separable. G. LEMMA. Let A be a primary and S a central separable R-algebra. Then for any two homomorphisms σ, σ : S A of R-algebras there is a unit u in A such that s σ u = s σ for s S. Here we call A primary if A/JA is a simple F -algebra. Proof. σ and σ induce homomorphisms S/JS A/JA; here A/JA is a simple F -algebra and S/JS is a central simple F -algebra. If the induced homomorphisms are conjugate by a unit of A/JA then lemma F implies that σ and σ are conjugate by a unit in A. Therefore we may assume that R = F and A is simple. In this case the lemma can be proves as in [9, (7.21)]. We not try to put things together. H. THEOREM. Let A be a crossed product of G over A 1 with automorphism system α, and set N := {g G α(g) Inn A 1 }. Let b e be a block of A N such that C b (A 1 )/(JZA 1 )C b (A 1 ) is a separable F -algebra and e ZA. Denote by S an inertial subalgebra of C b (A 1 ) and by Q a maximal central separable R-subalgebra of S. Then the following holds: (i) ea is a crossed product of G/N over b. (ii) The map C ea (Q) R Q ea, c q cq, is an isomorphism of R-algebras. (iii) C ea (Q) is a crossed product of G/N over C b (Q). (iv) The map ea 1 eza1 S C S (Q) C b (Q), x c xc, is an isomorphism of R-algebras. (v) C S (Q) is a separable local R-algebra. (vi) For any transversal T for N in G there are units u(g) in C b (A 1 ) such that the map α : G/N Aut C b (Q) with (xc) α (gn) = x α(g) c gu(g) for x ea 1, c C S (Q), g T is an automorphism system for C ea (Q). Proof. Obviously A is a crossed product of G/N over A N. Since e is central in A, (i) holds. (ii) is an immediate consequence of lemma E. By theorem C, the map A 1 ZA1 C AN (A 1 ) A N, x c xc, is an isomorphism of R-algebras. Therefore the map ea 1 eza1 C b (A 1 ) b, x c xc, is an isomorphism, too. By theorem C, C AN (A 1 ) is free as a module over ZA 1, so C b (A 1 ) is projective as a module over eza 1. But eza 1 is local since b is a block ideal, so C b (A 1 ) is free over eza 1, and we may apply lemma D to C b (A 1 ) and eza 1 to get an isomorphism eza 1 eza1 S S C b (A 1 ), z s zs. This implies that the map ea 1 eza1 S S b, x s xs, is an isomorphism. In particular, S has exactly one block. Since S is separable this implies that S is primary. The group G acts on C A (A 1 ), so it also acts on C AN (A 1 ) and on C b (A 1 ). For any element g G, S g is another inertial subalgebra of C b (A 1 ). By Azumaya s theorem [1] there exists a unit v(g) of C b (A 1 ) such that S gv(g) = S. Then by lemma G we can find a unit w(g) of S such that q gv(g)w(g) = q for q Q. This shows that we may apply lemma B to get that C ea (Q) is a crossed product of G/N over C b (Q), and we have proved (iii).

6 Communications in Algebra 13(1), (1985) 6 By lemma E the map C S (Q) R Q S, c q cq, is an isomorphism. Therefore b = ea 1 eza1 S C S (Q) R Q. On the other hand, b = C b (Q) R Q, again by lemma E. This implies (iv). Since S is primary and C S (Q) R Q = S, C S (Q) is primary, too. Since Q is a maximal central separable R-subalgebra of S, C S (Q) contains only trivial idempotents, so C S (Q) is local. The separability of S implies that of C S (Q), so (v) is proved. Now choose a transversal T for N in G and units u g of A in A g such that xu g for x A 1 and g T. Then the elements eu g v(g)w(g) are units in C ea (Q) contained in ea gn satisfying (xc) eugv(g)w(g) = x α(g) c gv(g)w(g) for x ea 1, c C S (Q), g T. This shows that (vi) is true. x α(g) = g It follows that apart from the two separable algebras Q and C S (Q) the structure of ea is determined by a crossed product of G/N over ea 1. So we have been able to remove the normal subgroup N. It is well-known that the hypothesis e ZA can be removed by first passing to the stabilizer of b e. I. THEOREM. Let A be a crossed product of G over A 1, and let b e be a block of A 1. Choose a transversal g 1,..., g t for G(b) in G and units u 1,..., u t of A in A g1,..., A gt, respectively. Then the following holds: (i) eae = ea G(b) is a crossed product of G(b) over b with g-component ea g = A g e for g G(b). (ii) If α is an automorphism system for A then its restriction is an automorphism system for eae. (iii) τg(b) G (e) is an idempotent in ZA with AeA = τ G(b) G (e)a. (iv) The maps and Mat(t, eae) AeA, [a ij ] t i,j=1 t i,j=1 i a ij u j AeA Mat(t, eae), x [eu i x j e] t i,j=1, are isomorphisms of R-algebras and inverse to each other. (v) The maps Z(eAe) Z(AeA), z τ G G(b) (z), and Z(AeA) Z(eAe), y ey, are isomorphisms of R-algebras and inverse to each other. Proof. If g and h are elements in G lying in different cosets of G(b) then e g and e h are different idempotents in ZA 1, so e g e h = 0. This implies ea g e = ea 1 u g e = ea 1 u g e g e = 0 for any element g G\G(b) and any unit u g of A in A g, and ea g = ea 1 u g = A 1 eu g = A 1 u g e g = A g e = ea g e for any element g G(b) and any unit u g of A in A g. Furthermore, e g eu g = ee g = e. Therefore eu g is a unit of eae in ea g, and (i) and (ii) are proved.

7 Communications in Algebra 13(1), (1985) 7 From e C A (A G(b) ) we conclude τg(b) G (e) C A(A G ) = ZA and τg(b) G (e) = t i eu i AeA. On the other hand, e = eτg(b) G (e), so AeA = Aτ G(b) G (e)a = i=1 τg(b) G (e)a. Since τ G(b) G (e) is a sum of pairwise orthogonal idempotents it is an idempotent, too. So (iii) follows. For elements a ij, b kl eae and x AeA we have i,j,k,l eu k i,j i i a ij u j k b klu l = i,j,k,l a ij u j = i,j,l = i,l l e = i,j i a ij u j e gj e g k k b klu l i a ij eb jl u l i j a ij b jl u l, u k e g k e gi i a ij u j e gj e g l l and i,j = ea kl e = a kl i eu i x j eu j = τg(b) G (e)xτ G(b) G (e) = x. This implies (iv). In particular, we get an isomorphism ZMat(t, eae) Z(AeA). When we multiply this map with the natural isomorphism Z(eAe) ZMat(t, eae) we get the first map in (v). Its inverse is given by the second map since eτg(b) G (z) = eτ G(b) G (ez) = ez = z for z Z(eAe). This completes the proof of theorem I. At this point it seems natural to introduce some splitting condition in order to avoid all the extensions of R. J. THEOREM. Let A be a crossed product of G over A 1 with automorphism system α, and set N := {g G α(g) Inn A 1 }. Let b e be a block of A N such that C b (A 1 )/(JZA 1 )C b (A 1 ) is a central simple F -algebra, and denote by S an inertial subalgebra of C b (A 1 ). Then the R-algebras τ G G(b) (e)a and Mat([G : G(b)], C R S) are isomorphic for a crossed product C of G(b)/N over ea 1 with automorphism system α : G(b)/N Aut (ea 1 ) satisfying x α (gn) = x α(g) for x ea 1, g T where T is an arbitrary transversal for N in G(b). Proof. An application of theorem I to the crossed product A of G/N over A N shows that the R-algebras τ G G(b) (e)a and Mat([G : G(b)], ea G(b)) are isomorphic. Furthermore, A G(b) is a crossed product of G(b) over A 1 satisfying the conditions of theorem H. Z(C b (A 1 )/(JZA 1 )C b (A 1 )) = F e + (JZA 1 )C b (A 1 ) implies ZS = Re, so S itself is a central separable R-algebra. Our result follows

8 Communications in Algebra 13(1), (1985) 8 from theorem H. 2. GROUP ALGEBRAS In order to apply theorem J to blocks with normal defect groups several other things have to proved. The following statement also occurs in [4]. Since our general hypothesis is different from that in [4] we cannot use the proof given there. K. LEMMA. For any natural number n not divisible by the characteristic of F, any commutative R-algebra A and any element x 1 + JA there is exactly one element y 1 + JA such that x = y n. Proof. Since A is complete and 1 + JA is closed in the topology defined by JA it is enough to show that for any k N there is a unique element y k + (JA) k A/(JA) k satisfying y n k + (JA) k = x + (JA) k and y k + JA = 1 + JA. This is straight forward by induction on k. We need lemma K to prove the following. L. PROPOSITION. Let G be a finite group such that G is not divisible by the characteristic of F, and let A be a crossed product of G over A 1. Then H n (G, 1 + JZA 1 ) = 0 for n N. Proof. Lemma K and [8, I.16.20]. In our setup a twisted group algebra of G over R is nothing but a crossed product of G over R. Thus we obtain: M. LEMMA. Let D be a finite p-group and G a p -subgroup of Aut D. Let A be a crossed product of G over the group algebra RD such that the automorphism system for A is induced by the action of G on D. Then A is a twisted group algebra of the semidirect product DG of D with G over R. Proof. We choose units u g of A in A g such that g du g = d g for d D, g G. Then h u 1 g u gh d gh u gu h = d h 1 g 1 gh = d for d D, g, h G, so gh u gu h A 1 C A (RD) = ZRD. The map (g, h) gh u gu h defines a 2-cocycle in Z 2 (G, UZRD). Since H 2 (G, UZRD) = H 2 (G, UR) H 2 (G, 1 + JZRD) = H 2 (G, UR) by proposition L this 2-cocycle is equivalent to a 2-cocycle with values in UR. Therefore we may assume gh u gu h UR for g, h G. But then the elements

9 Communications in Algebra 13(1), (1985) 9 du g (d D, g G) are a basis for A such that (cu g )(du h ) = cd g 1 u g u h (UR)cd g 1 u gh for elements c, d D, g, h G. The result follows. We also need a result on the behavior of defect groups with respect to normal subgroups. A proof of these results can be obtained from [4]. Note however, that we do not have any splitting hypothesis on R. N. PROPOSITION. Let B E be a block of the group algebra RG, and let b e be a block of RN with Ee 0 where N is a normal subgroup of G. Then Ee is a block idempotent in RG(b), any defect group D of the corresponding block is a defect group of B E, and D N is a defect group of b e. We are now in a position to prove theorem A. Proof of theorem A. By proposition N, Ee is a block idempotent in RG(b), and the corresponding block has defect group D. Since D is normal in G we have ZRG = (ZRG RC G (D)) + JZRG, so E is contained in RC G (D). This implies Ee = e, and E = τg(b) G (e) by theorem I. We choose an element c C RG(b) (D) with e = τ G(b) D (c). Since e RC G(D) we may assume c RC G (D). After replacing c by ece we may also assume c b. Then τ DC G(D) D (c) = τ C G(D) ZD (c) Zb so τ DC G(D) D (c) αe (mod JZRC G (D)) for some α R. This implies e = τ G(b) D G(b) (c) τdc G (D) (αe) [G(b) : DC G(D)]αe (mod JZRG(b)), so [G(b) : DC G (D)] is not divisible by p. Thus DC G (D)/C G (D) is a Sylow p- subgroup of G(b)/C G (D), so possesses a complement H by the Schur-Zassenhaus theorem. We regard RG as a crossed product of G/D over RD. The elements of H induce p -automorphisms on F D where as the elements in URD induce p- automorphisms on F D by conjugation. Therefore the normal subgroup N of G/D inducing inner automorphisms of RD must be DC G (D)/D. C RDCG (D)(RD) is a crossed product of DC G (D)/D over ZRD by theorem C. Thus C RDCG (D)(RD)/(JZRD)C RDCG (D)(RD) is a crossed product of DC G (D)/D over F. Since C G (D) C RDCG (D)(RD) we even get C RDCG (D)(RD)/(JZRD)C RDCG (D)(RD) = F [DC G (D)/D]. Obviously the image of e in F [DC G (D)/D] is a block idempotent of defect zero, so C b (RD)/(JZRD)C b (RD) is a central simple F -algebra. An application of theorem J shows that B is isomorphic to C R S for some central separable R-algebra S and some crossed product C of H over erd. Since erd = RD our result follows from lemma M.

10 Communications in Algebra 13(1), (1985) 10 REFERENCES 1. G. AZUMAYA, On maximally central algebras, Nagoya Math. J. 2 (1951), H. BASS, Lectures on topics in algebraic K-theory, Tata Institute, Bombay, R. BRAUER, Zur Darstellungstheorie der Gruppen endlicher Ordnung I, Math. Z. 63 (1956), E.C. DADE, Block Extensions, Illinois J. Math. 17 (1973), E.C. DADE, Group-graded rings and modules, Math. Z. 174 (1980), F. DEMEYER and E. INGRAHAM, Separable algebras over commutative rings, Lect. Notes in Math. Vol. 181, Springer-Verlag, Berlin J.A. GREEN, Axiomatic representation theory for finite groups, J. Pure Applied Algebra 1 (1971), B. HUPPERT, Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg- New York I. REINER, Maximal orders, Academic Press, London-New York-San Francisco, W.F. REYNOLDS, Blocks and normal subgroups of finite groups, Nagoya Math. J. 22 (1963), 15-32